Problem 21 - College Physics

Question

\(\bullet\) \(\bullet\) A steel cable with cross-sectional area of 3.00 \(\mathrm{cm}^{2}\) has an elastic limit of \(2.40 \times 10^{8}\) Pa. Find the maximum upward acceleration that can be given to a 1200 kg elevator supported by the cable if the stress is not to exceed one-third of the elastic limit.

Step-by-Step Solution

Answer

The maximum upward acceleration is \( 10.19\, \text{m/s}^2 \).

Step 1: Understanding Stress and Elastic Limit

The elastic limit represents the maximum stress that a material can withstand without undergoing permanent deformation. We need to ensure that the stress on the cable does not exceed one-third of its elastic limit. The elastic limit of the cable is given as \(2.40 \times 10^8\, \text{Pa}\). Therefore, the allowed stress is one-third of this limit, which is \[\text{Allowed Stress} = \frac{2.40 \times 10^8}{3}\, \text{Pa}\].

Step 2: Calculate Allowed Stress

Calculate one-third of the elastic limit to determine the maximum allowed stress on the cable: \[ \text{Allowed Stress} = \frac{2.40 \times 10^8}{3} = 0.80 \times 10^8\, \text{Pa}\]. This is the maximum stress the cable can handle.

Step 3: Use Stress Formula for Maximum Force

Stress is defined as force per unit area. We have the stress value and the cross-sectional area of the cable, so we can find the force: \[ \text{Stress} = \frac{F}{A} \implies F = \text{Stress} \times A \]. Substitute the values: \[ F = 0.80 \times 10^8\, \text{Pa} \times 3.00 \times 10^{-4}\, \text{m}^2 = 24000\, \text{N} \]. This is the maximum force the cable can exert.

Step 4: Apply Newton's Second Law to Find Maximum Acceleration

We need to find the maximum upward acceleration for the elevator. The total force acting on the elevator includes the maximum force from the cable and the gravitational force: \[ F - mg = ma \], where \( m = 1200\, \text{kg} \), \( g = 9.81\, \text{m/s}^2 \), and \( \text{Allowed } F = 24000\, \text{N} \). Solve for \( a \): \[ 24000 - 1200 \times 9.81 = 1200a \], which simplifies to \[ a = \frac{24000 - 11772}{1200} \].

Step 5: Calculate Maximum Acceleration

Calculate the result from the previous equation: \[ a = \frac{12228}{1200} = 10.19\, \text{m/s}^2 \]. Therefore, the maximum upward acceleration that the elevator can have is \( 10.19\, \text{m/s}^2 \).

Key Concepts

Stress and Elasticity

Stress is a fascinating concept in physics because it describes the internal resistance of a material when subjected to external forces. Understanding stress is critical when dealing with materials, as every material has a limit to how much stress it can endure before it changes permanently. This point is known as the elastic limit. Specifically, for the steel cable in the problem, this limit is given as \(2.40 \times 10^8\, \text{Pa}\). This means that if the stress goes beyond this value, the cable might not return to its original shape once the force is removed.
However, to ensure safety and dependability, engineers often design systems to operate well below this elastic limit. In the given exercise, we impose a restriction where the stress should not exceed one-third of the cable's elastic limit. Doing this helps in prolonging the life of the material and ensuring safety.

• The allowed stress = \(\frac{2.40 \times 10^8}{3}\, \text{Pa}\).
• This precautionary measure guarantees the structural integrity of the elevator's support system.

Newton's Second Law

Newton's Second Law is a cornerstone of dynamics and relates to how objects interact with forces and motion. In simple terms, it states that the force acting on an object is equal to the mass of that object multiplied by its acceleration. The law is typically written as \( F = ma \), where \( F \) is the force applied, \( m \) is the mass, and \( a \) is the acceleration.
In the elevator problem, applying Newton's Second Law assists in finding the maximum acceleration that the elevator can achieve. The total force includes both the force the cable can exert and the force due to gravity acting on the elevator itself.

• Gravity always acts downwards, making it a constant factor in the force equation.
• The balance of the upward force from the cable and the downward gravitational force dictates the possible acceleration.

By calculating these forces, one can isolate and solve for the desired maximum acceleration, ensuring that the system operates optimally within safe limits.

Maximum Acceleration Calculation

Determining the maximum acceleration involves applying the principles of stress and Newton's Second Law to set parameters for safe operation. First, calculate the maximum allowable force that the cable can sustain without surpassing its safe stress limit. This force is found by multiplying the allowed stress by the cross-sectional area of the cable:

• \( F = \text{Allowed Stress} \times A \)
• \( F = 0.80 \times 10^8\, \text{Pa} \times 3.00 \times 10^{-4}\, \text{m}^2 = 24000\, \text{N}\)

The force from the cable provides part of the equation to solve for acceleration. Using the rearranged formula of Newton's Second Law:

• Net force = Total force - gravitational force
• \( 24000 - 1200 \times 9.81 = 1200a \)

Solving this equation gives:

• \( a = \frac{24000 - 11772}{1200} = 10.19\, \text{m/s}^2 \)
• This value represents the maximum safe upward acceleration for the elevator.

This detailed calculation ensures that mechanical components like the steel cable are not overloaded, maintaining the functionality and safety of the system.